Wavelet-Galerkin solution of boundary value problems

In this paper we review the application of wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of wavelets is described here using the language and mathematics of signal processing. We show a method of adapting wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.